Kourovka Notebook Problem 21.125

Definitions and mathematical statement

Let $F_m$ be the free group of rank $m\ge1$ with basis $x_1,\ldots,x_m$, and let $|g|$ be reduced word length in that basis. Suppose $\varphi\in\operatorname{Aut}(F_m)$ satisfies \[ c n^{m-1}\le\max_i|\varphi^n(x_i)|\le C n^{m-1} \quad(n\ge n_0) \] for some $c,C>0$ and $n_0\ge1$. This specifies polynomial growth of degree $m-1$ as in the source. Define the free-by-cyclic group \[ G_\varphi=\langle x_1,\ldots,x_m,t\mid t^{-1}x_it=\varphi(x_i)\ (1\le i\le m)\rangle. \]

For completeness, a cube complex is formed by gluing faces of unit Euclidean cubes by isometries. A Salvetti complex is the union of coordinate tori indexed by the finite cliques of a graph, with one common vertex and one generator loop per graph vertex. A cube complex is special if it admits a cubical local isometry to such a Salvetti complex. On vertex links this means an injective map onto a full subcomplex; full means that every simplex whose vertices lie in the image also lies in the image. A group is virtually special if some finite-index subgroup is isomorphic to the fundamental group of a special cube complex.

(a) Is $G_\varphi$ virtually special for every automorphism satisfying the displayed growth hypothesis?

(b) In particular, is every Hydra group \[ G_m=\langle a_1,\ldots,a_m,t\mid t^{-1}a_1t=a_1,\quad t^{-1}a_it=a_ia_{i-1}\ (2\le i\le m)\rangle \] virtually special? The target is existence of some finite-index subgroup with the stated cubical realization. No bound on its index, dimension, or number of cubes is required in either part.

Short English statement

Are free-by-cyclic groups with maximal-degree polynomial monodromy virtually special, including all the explicitly presented Hydra groups?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2634 (KOU-21.125), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.125. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.
• S3 N. Brady and I. Soroko, Dehn functions of subgroups of right-angled Artin groups, Theorem 4.12; local-isometry criterion for special cube complexes. https://sites.math.unt.edu/~soroko/Dehn-functions-RAAGs.pdf.